On Lipschitz Spaces, Semigroup Transference, and Heat Kernel Derivative Estimates in Dunkl--Schrödinger Analysis
Abstract
In this paper, we study Lipschitz spaces adapted to Dunkl--Schrödinger operators ℒ = −Δk + V, where Δk is the Dunkl Laplacian and V ≥ 0 belongs to the reverse Hölder class RHq(dw) with q > max(1, N/2). For 0 < β< 2, we establish the full equivalence between the heat semigroup-based Lipschitz spaces Λ̃ℒ,kβ/2(ℝN) and the pointwise weighted Zygmund classes Λℒ,kβ(ℝN) defined via the critical radius function m(x). Remarkably, this characterization shows that the spatial smoothness component decouples completely from the underlying reflection group symmetries and root systems. As an intermediate result, we establish new equivalences for inhomogeneous weighted Lipschitz spaces associated with the unperturbed Dunkl Laplacian Δk. A central difficulty in this context --- arising from the necessity to disentangle the Euclidean distance from the orbit metric d(x,y) --- is resolved through a detailed geometric analysis. Finally, we provide pointwise Gaussian upper bounds for the time derivatives of the Dunkl--Schrödinger heat kernel, which are of independent interest for future work in harmonic analysis on Dunkl structures. Crucially, we do not assume the potential V to be G-invariant under the action of the Weyl group, which introduces major geometric difficulties resolved here via a local analysis of the orbit metric.
Related
- Characterization of sets in metric spaces where the big Lipschitz derivative is infinite
- Semigroup properties of Schrödinger operators with complex singular potentials
- Global well-posedness for the derivative nonlinear Schrödinger equation on the circle
- Explicit quasi-periodic solutions to the discrete derivative nonlinear Schrödinger equation
- Littlewood-Paley and Carleson measure characterizations of Lipschitz spaces adapted to Schrödinger operators
BibTeX
@misc{hejnalyzwa2026lipschitz,
title = {{On Lipschitz Spaces, Semigroup Transference, and Heat Kernel Derivative Estimates in Dunkl--Schrödinger Analysis}},
author = {Agnieszka Hejna-Łyżwa},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.27714},
eprint = {2608.27714},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.27714},
}
RIS
TY - UNPB
TI - On Lipschitz Spaces, Semigroup Transference, and Heat Kernel Derivative Estimates in Dunkl--Schrödinger Analysis
AU - Agnieszka Hejna-Łyżwa
PY - 2026
DA - 2026/08/27
PB - arXiv
DO - 10.48550/arxiv.2608.27714
UR - https://arxiv.org/abs/2608.27714
AB - In this paper, we study Lipschitz spaces adapted to Dunkl--Schrödinger operators ℒ = −Δ_k + V, where Δ_k is the Dunkl Laplacian and V ≥ 0 belongs to the reverse Hölder class RH^q(dw) with q > max(1, N/2). For 0 < β< 2, we establish the full equivalence between the heat semigroup-based Lipschitz spaces Λ̃_ℒ,k^β/2(ℝ^N) and the pointwise weighted Zygmund classes Λ_ℒ,k^β(ℝ^N) defined via the critical radius function m(x). Remarkably, this characterization shows that the spatial smoothness component decouples completely from the underlying reflection group symmetries and root systems. As an intermediate result, we establish new equivalences for inhomogeneous weighted Lipschitz spaces associated with the unperturbed Dunkl Laplacian Δ_k. A central difficulty in this context --- arising from the necessity to disentangle the Euclidean distance from the orbit metric d(x,y) --- is resolved through a detailed geometric analysis. Finally, we provide pointwise Gaussian upper bounds for the time derivatives of the Dunkl--Schrödinger heat kernel, which are of independent interest for future work in harmonic analysis on Dunkl structures. Crucially, we do not assume the potential V to be G-invariant under the action of the Weyl group, which introduces major geometric difficulties resolved here via a local analysis of the orbit metric.
ER -